Effects of viscous dissipation on the heat transfer in a forced pipe flow. Part 2: Thermally developing flow

In this part of the study, consideration is given to thermally developing laminar forced convection in a pipe including viscous dissipation. The axial heat conduction in the fluid is neglected. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Both the wall heating (the fluid is heated) case and the wall cooling (the fluid is cooled) case are considered. The distributions for the developing temperature and local Nusselt number in the entrance region are obtained. Results show that the temperature profiles and local Nusselt number are influenced by the Brinkman number (Br) and the thermal boundary condition used for the wall. Significant viscous dissipation effects have been observed for large Br.     

Three-dimensional laminar slip-flow and heat transfer in a rectangular microchannel with constant wall temperature

Three-dimensional laminar slip-flow and heat transfer in rectangular microchannels having constant temperature walls are studied numerically using the finite-volume method for thermally and simultaneously developing flows. The Navier–Stokes and energy equations are solved with velocity slip and temperature jump at the wall. A modified convection–diffusion coefficient at the wall–fluid interface is defined to incorporate the temperature-jump boundary condition. Validity of the numerical simulation procedure is established and the effect of rarefaction on hydrodynamicaly developing flow field, pressure gradient and entrance length is analyzed. A correlation for the fully developed friction factor is presented as a function of Knudsen number (Kn) and aspect ratio (α). The influence of rarefaction on the Nusselt (Nu) number is investigated for thermally and simultaneously developing flows. The effect of velocity slip is found to increase the Nu number, while the temperature-jump tends to decrease it, and the combined effect could result in an increase or a decrease in the Nu number. In the fully developed region, there could be high as 15% increase or low as 50% decrease in Nu number is plausible for the range of parameters considered in this work.     

Forced Convection Heat Transfer Simulation Using Dissipative Particle Dynamics

Dissipative particle dynamics (DPD) with energy conservation was applied to simulate forced convection in parallel-plate channels with boundary conditions of constant wall temperature (CWT) and constant wall heat flux (CHF). DPD is a coarse-grained version of molecular dynamics. An additional equation for energy conservation was solved along with conventional DPD equations, where inter-particle heat flux accounts for changes in mechanical and internal energies when particles interact with surrounding particles. The solution domain was considered to be two–dimensional with periodic boundary condition in the flow direction and additional layers of particles on the top and bottom of the channel to apply no-slip and wall temperature boundary conditions. The governing equation for energy conservation was modified based on periodic fully developed velocity and temperature conditions. The results were shown via velocity and temperature profiles across the channel cross-section. The Nusselt numbers for CWT and CHF were calculated from the temperature gradient at the wall using a second order accurate forward difference approximation. The results agreed well with the exact solutions to within 2.3%.     

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer rates, as given by the Nusselt number, is numerically evaluated subject to constant wall heat flux (H2) and constant wall temperature (T) thermal boundary conditions. Numerical results are obtained using a continuum based, three-dimensional, compressible, unsteady computational fluid dynamics algorithm with slip velocity and temperature jump boundary conditions applied to the momentum and energy equations, respectively. For the limiting case of parallel plate channels, analytic solutions for the thermally and hydrodynamically fully developed momentum and energy equations are derived, subject to both first- and second-order slip velocity and temperature jump boundary conditions, from which analytic Nusselt number solutions are then obtained. Excellent agreement between the analytical and numerical results verifies the accuracy of the numerical algorithm, which is then employed to obtain three-dimensional rectangular channel and thermally/hydrodynamically developing Nusselt numbers. Nusselt number data are presented as functions of Knudsen number, Brinkman number, Peclet number, momentum and thermal accommodation coefficients, and aspect ratio. Rarefaction and viscous dissipation effects are shown to significantly affect the convective heat transfer rate in the slip flow regime.     

Second law analysis of fully developed laminar flow for rectangular ducts with semicircular ends

Second law analysis is performed analytically for rectangular ducts with semicircular ends in laminar flow. Two different situations are considered for the analysis. In the first case, boundaries of duct are considered in constant wall temperature. In the second case, constant wall heat flux boundary conditions applied. Entropy generation is obtained for various cross sectional areas, various wall heat flux and Reynolds numbers. It is found that with the increasing aspect ratio (β) values for both constant wall temperature (CWT) and constant heat flux (CHF) total entropy generation increases, however, required pumping power also increases.     

Analysis of laminar heat transfer in micro-Poiseuille flow

The present study examines laminar forced convective heat transfer of a Newtonian fluid in a microchannel between two parallel plates analytically. The viscous dissipation effect, the velocity slip and the temperature jump at the wall are included in the analysis. Both hydrodynamically and thermally fully developed flow case is examined. Either the hot wall or the cold wall case is considered for the two different thermal boundary conditions, namely the constant heat flux (CHF) and the constant wall temperature (CWT). The interactive effects of the Brinkman number and the Knudsen number on the Nusselt numbers are analytically determined. Different definitions of the Brinkman number based on the definition of the dimensionless temperature are discussed. It is disclosed that for the cases studied here, singularities for the Brinkman number-dependence of the Nusselt number are observed and they are discussed in view of the energy balance.     

Mixed convection about a rotating sphereConvection mixte autour d'une sphere tournanteMischkonvektion an einer rotierenden kugelCмeшaHHaя кoHвeкция oкoлo вpaщaющeйCя Cфepы

The paper presents a theoretical analysis of flow and heat transfer characteristics of the effects of buoyancy force on laminar boundary layer over a rotating sphere in forced flow under two kinds of heating conditions: uniform wall temperature and uniform surface heat flux. By applying appropriate coordinate transformations and using Merk's types of series, the governing momentum and energy equations are reduced to a set of coupled ordinary differential equations, which depend on wedge, rotation and buoyancy parameters. Numerical computations are carried out for Prandtl numbers 0.7,1.0 and for various values of buoyancy and rotation parameters. For aiding flow, it is found that both the friction factor and the local Nusselt number increase with increasing buoyancy force. The local free stream velocity increases with buoyancy which, in turn, affects the friction coefficient and Nusselt number. The coupling between rotation and buoyancy results in increased overshooting of the velocity profiles in the vicinity of the rotating sphere. For an equivalent buoyancy effect, heating by uniform surface heat flux yields larger local Nusselt number than heating by uniform wall temperature. The ratio Nu_UHF/Nu_UWT is higher for the rotating sphere (as compared to a nonrotating case) and further the ratio increases as the sphere spins faster. The effect of free stream, rotation and buoyancy on the eruption of flow is examined and also a suggestion for further investigation is made.     

Heat and fluid flow characteristics of gases in micropipes

In this study, laminar forced convective heat transfer of a Newtonian fluid in a micropipe is analyzed by taking the viscous dissipation effect, the velocity slip and the temperature jump at the wall into account. Hydrodynamically and thermally fully developed flow case is examined. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Either wall heating (the fluid is heated) case or wall cooling (the fluid is cooled) case is examined. The Nusselt numbers are analytically determined as a function of the Brinkman number and the Knudsen number. Different definitions of the Brinkman number based on the definition of the dimensionless temperature are discussed. It is disclosed that for the cases studied here, singularities for the Brinkman number-dependence of the Nusselt number are observed and they are discussed in view of the energy balance.     

Laminar natural convection heat and mass transfer in vertical rectangular ducts

The present work investigates numerically the laminar natural convection heat and mass transfer in open vertical rectangular ducts with uniform wall temperature/uniform wall concentration (UWT/UWC) or uniform heat flux/uniform mass flux (UHF/UMF) boundary conditions. The vorticity–velocity formulation is applied to solve for the coupled momentum, energy and concentration equations. Results of dimensionless induced volume rate Q, average Nusselt number Nu and Sherwood number Sh are presented in terms of channel length L, buoyancy ratio N, Grashof number Gr, Schmidt number Sc and aspect ratio γ. Analytical solutions for Q, Nu and Sh for the UWT/UWC case are derived under fully developed condition. In addition, the correlation equations of Q, Nu and Sh for both boundary conditions are also presented.     

Fully developed forced convective heat transfer in an annulus partially filled with metallic foams: An analytical solution

An analytical solution for fully developed forced convective heat transfer in an annulus partially filled with metallic foam was proposed. The inner surface attached with an annular metallic foam layer was exposed to constant heat flux while the outer surface was adiabatic. In the metallic foam region, the Brinkman–Darcy equation was used to describe the fluid flow and the thermal non-equilibrium model was employed to establish the heat transfer equations. At the porous-fluid interface, no-slip coupling conditions were utilized to couple flow and heat transfer of the porous and open regions. A closed-form analytical solution was obtained for velocity and temperature profiles. The explicit form of friction factor and the Nusselt (Nu) number were also provided. The solutions were validated by two extreme cases: the empty annulus and the annulus fully filled with metallic foam. The effects of key parameters on friction factor, Nu number, and j/f^1/3 were examined. The relationship between flow heterogeneity and heat transfer was also discussed by introducing the flow heterogeneity coefficient. The porosity, pore density, and foam thickness for engineering applications were recommended. In the present analytical solution, a benchmark was also established for improving discretizing schemes in numerical works.     

Numerical solution of power law fluids flow and heat transfer with a magnetic field in a rectangular duct

The steady laminar flow and heat transfer of an incompressible, electrically conducting, power law non-Newtonian fluids in a rectangular duct are studied in the presence of an external uniform magnetic field. The momentum and energy equations are solved iteratively using a finite difference method. Two cases of the thermal boundary conditions are considered; (1) T thermal boundary condition “constant temperature at the wall” and (2) H2 thermal boundary condition “constant heat flux at the wall”. The viscous and Joule dissipations are taken into consideration in the energy equation. A numerical solution for the governing partial differential equations is developed and the influence of the magnetic field on the velocity distribution, the friction factor and the average Nusselt number are discussed.     

Conjugate forced convection heat transfer from a flat plate by laminar plane wall jet flow

An analytical solution is investigated for forced convection heat transfer from a laminar plane wall jet as conjugate case. For Re ? 1, boundary layer theory is used for the investigation. The problem has been solved for two classic cases such as Pr ? 1 and Pr ? 1. The conjugate model consists of considering the full Navier-Stokes equation in the fluid medium and coupling of energy equations in the fluid and the slab through the interface boundary conditions. Closed-form relations are found for Nusselt number (Nu), average Nusselt number and conjugate interface boundary temperature (θ_b). The effects of the Reynolds number (Re), the Prandtl number (Pr), the thermal conductivity ratio (k) between the slab and the fluid medium and the slab aspect ratio (λ) are investigated on the heat transfer characteristics. The analytical results are compared with the full numerical results.     

Numerical study of compressible convective heat transfer with variations in all fluid properties

The effects of property variations in single-phase laminar forced micro-convection with constant wall heat flux boundary condition are investigated in this work. The fully-developed flow through micro-sized circular (axisymmetric) geometry is numerically studied using two-dimensional continuum-based conservation equations. The non-dimensional governing equations show significance of momentum transport in radial direction due to μ(T) variation and energy transport by fluid conduction due to k(T) variation. For the case of heated air, variation in C_p(T) and k(T) causes increase in Nu. This is owing to: (i) reduction in T_w, (T_w ? T_m), and (?T/?r)_w and (ii) change in ?T_m/?z results in axial conduction along the flow. The effects of ρ(p,T) and μ(T) variation on convective-flow are indirect and lead to: (i) induce radial velocity which alters u(r) profile significantly and (ii) change in (?u/?r)_w along the flow. It is proposed that the deviation in convection with C_p(T), k(T) variation is significant through temperature field than ρ(p,T), μ(T) variation on velocity field. It is noted that Nu due to variation in properties differ from invariant properties (Nu = 48/11) for low subsonic flow.     

Numerical study on coupled fluid flow and heat transfer process in parabolic trough solar collector tube

A unified two-dimensional numerical model was developed for the coupled heat transfer process in parabolic solar collector tube, which includes nature convection, forced convection, heat conduction and fluid-solid conjugate problem. The effects of Rayleigh number (Ra), tube diameter ratio and thermal conductivity of the tube wall on the heat transfer and fluid flow performance were numerically analyzed. The distributions of flow field, temperature field, local Nu and local temperature gradient were examined. The results show that when Ra is larger than 10⁵, the effects of nature convection must be taken into account. With the increase of tube diameter ratio, the Nusselt number in inner tube (Nu₁) increases and the Nusselt number in annuli space (Nu₂) decreases. With the increase of tube wall thermal conductivity, Nu₁ decreases and Nu₂ increases. When thermal conductivity is larger than 200 W/(m K), it would have little effects on Nu and average temperatures. Due to the effect of the nature convection, along the circumferential direction (from top to down), the temperature in the cross-section decreases and the temperature gradient on inner tube surface increases at first. Then, the temperature and temperature gradients would present a converse variation at θ near π. The local Nu on inner tube outer surface increases along circumferential direction until it reaches a maximum value then it decreases again.     

Analysis of wind flow around a parabolic collector (2) heat transfer from receiver tube

Parabolic collectors of commercial solar thermal power plants are subject to variable convection heat transfer from the receiver tube. In the present study heat transfer from a receiver tube of the parabolic trough collector of the 250 kW solar power plants in Shiraz, Iran, is studied taking into account the effects of variation of collector angel of attack, wind velocity and its distribution with respect to height from the ground.The governing equations for the two-dimensional steady state wind flow include continuity, momentum and energy equations and RNG-based k–ε model for turbulence scheme. Finite volume discretization method is used to solve the governing equations with wall function boundary condition and the SIMPLE approach is employed to iterate for the pressure correction and convergence of the velocity field. The momentum equation contains buoyancy force when the buoyancy effect is high and force convection effect is low.Computation is carried out for various wind velocities and different collector orientations with respect to wind direction. For solution of the energy equation, temperature of the receiver tube is taken as 350 K and ambient temperature is assumed to be 300 K. Various recirculation and temperature fields were observed around the receiver tube for different flow conditions. Effect of collector orientation on the average Nu number for the receiver tube was found negligible when the wind speed is low (Re⩽4.5×10⁵ based on the collector aperture). But when the wind velocity is high (Re>4.5×10⁵), the collector effect on the variation of Nu around the glass cover of the absorber tube is considerable.     

Heat transfer of MHD flow over a wedge with a surface of mutable temperature

The objective of this study is to investigate the thermal distribution and heat transfer in the boundary layer of a wedge with a variable surface temperature in the presence of a magnetic field. To achieve this, we first used similarity solutions to transform the governing equations of magnetohydrodynamic flow for variable surface temperature conditions into ordinary differential equations. We then solved the resulting equations using the collocation method (CM) with different intensity magnetic fields and varying Hartmann (Ha) numbers and surface temperatures. The CM was further modified by incorporating boundary conditions. The results obtained from the solved equations were validated and compared with those obtained using the numerical Runge–Kutta fourth-order method and previous literature. Finally, we investigated the impact of various parameters on the friction coefficient (C_f) and Nusselt number (Nu), including the power of variable surface temperature (n), Prandtl (Pr) number, Eckert (Ec) number, the half angle of the wedge (φ), and Ha number. We considered values for these parameters within the ranges 0.5 ≤ n ≤ 1.5, 0.5 ≤ Pr ≤ 5, 0.001 ≤ Ec ≤ 0.002, 15° ≤ φ ≤ 60°, and 0 ≤ Ha ≤ 3. Our findings indicate that the slope of the boundary layer increases with increasing Ha or φ, resulting in an increase in C_f on the surface by up to 526%. The Nu number, calculated using the energy equation, increases up to 91.7%, 39.8%, and 1.43% with increasing Ha, n, and Ec, respectively, resulting in faster growth of the thermal boundary layer, which causes the thickness to decrease and the Nu number to rise. However, as φ increases, the Nu number drops on the surface, and the heat transfer behavior remains similar to that observed previously.     

Dual solutions in a double-diffusive convection near stagnation point region over a stretching vertical surface

The development of double-diffusive convection near stagnation point region over a stretching vertical surface with constant wall temperature has been investigated. The external flow and the stretching velocities are assumed to vary with $\sqrt{x}\text{,}$ where x is the distance from the slot where the stretching surface is issued. Using the local similarity method, it has been shown that a set of suitable similarity transformations reduces the non-linear coupled partial differential equations governing the flow, thermal and concentration fields into a set of non-linear coupled ordinary differential equations. The non-linear self-similar equations along with the boundary conditions form a two point boundary value problem and are solved using Shooting method, by converting into an initial value problem. In this method, the system of equations is converted into the set of first order system which is solved by fourth-order Runge–Kutta method. Flows with both assisting and opposing buoyancy forces are considered in the present investigation. The study reveals that the dual solutions of velocity, temperature and concentration exist for certain values of suction/injection and buoyancy parameters. Prandtl and Schmidt numbers strongly affect the thermal and concentration boundary layer thicknesses, respectively. The effects of various parameters on the velocity, temperature and concentration profiles are also presented here.     

Analysis of microchannel heat sinks for electronics cooling

This paper presents an analytical and numerical study on the heat transfer characteristics of forced convection across a microchannel heat sink. Two analytical approaches are used: the porous medium model and the fin approach. In the porous medium approach, the modified Darcy equation for the fluid and the two-equation model for heat transfer between the solid and fluid phases are employed. Firstly, the effects of channel aspect ratio (α_s) and effective thermal conductivity ratio (k?) on the overall Nusselt number of the heat sink are studied in detail. The predictions from the two approaches both show that the overall Nusselt number (Nu) increases as α_s is increased and decreases with increasing k?. However, the results also reveal that there exists significant difference between the two approaches for both the temperature distributions and overall Nusselt numbers, and the discrepancy becomes larger as either α_s or k? is increased. It is suggested that this discrepancy can be attributed to the indispensable assumption of uniform fluid temperature in the direction normal to the coolant flow invoked in the fin approach. The effect of porosity (ε) on the thermal performance of the microchannel is subsequently examined. It is found that whereas the porous medium model predicts the existence of an optimal porosity for the microchannel heat sink, the fin approach predicts that the heat transfer capability of the heat sink increases monotonically with the porosity. The effect of turbulent heat transfer within the microchannel is next studied, and it is found that turbulent heat transfer results in a decreased optimal porosity in comparison with that for the laminar flow. A new concept of microchannel cooling in combination with microheat pipes is proposed, and the enhancement in heat transfer due to the heat pipes is estimated. Finally, two-dimensional numerical calculations are conducted for both constant heat flux and constant wall temperature conditions to check the accuracy of analytical solutions and to examine the effect of different boundary conditions on the overall heat transfer.     

Dual solutions in a thermal diffusive flow over a stretching sheet with variable thickness

The development of thermal diffusive flow over a stretching sheet with variable thickness has been investigated. The non-linear coupled partial differential equations governing the flow and thermal fields are first transformed into a set of non-linear coupled ordinary differential equations by a set of suitable similarity transformations. The resulting system of coupled non-linear differential equations is solved using the Shooting method by converting into an initial value problem. In this method, the system of equations is converted into the set of first order system which is solved by fourth-order Runge–Kutta method. It is interesting to note that multiple solutions are observed for certain wall thickness parameter (β) and velocity power index (m). Velocity overshoot near the wall is observed for certain solution branches. The significant impacts on the boundary layer development along the wall on the velocity profiles and on the shear stress distribution in the fluid have been found by the non-flatness of the stretching surface. The mass suction effect is introduced by the non-flatness, when the velocity power index is less than one. The mass injection effect is lead to non-flatness when the velocity power index is greater than one. It is found that dual solution exists only for negative value of velocity power index (m). The presence of dual solutions in velocity and temperature fields for certain values of wall thickness parameter (β) and velocity power index (m) are revealed by this study.     

Development of free convection flow of a gas in a heated vertical open tube

The system is a vertical tube open at both ends and heated at the wall. An ambient gas (Pr = 0.7) enters the bottom of the tube with uniform velocity and temperature and flows up through the tube due to natural convection. The flow is assumed to be both stable and laminar. The incompressible thermal boundary layer equations for this situation were solved by a finite difference method for conditions of constant wall temperature and constant wall heat flux.